1201.4709 (Jeremy Quastel et al.)

Regularity and continuum statistics of the Airy1 process    [PDF]

Jeremy Quastel, Daniel Remenik

We obtain a formula for the $n$-dimensional distributions of the Airy$_1$ process in terms of a Fredholm determinant on $L^2(\rr)$, as opposed to the standard formula which involves extended kernels, on $L^2({1,...,n}\times\rr)$. The formula is analogous to an earlier formula of [PS02] for the Airy$_2$ process. Using this formula we are able to prove that the Airy$_1$ process is H\"older continuous with exponent $\frac12-$, and we explain how the same proof can be used to obtain the analogous result for the Airy$_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the Airy$_1$ process, analogous to that obtained in [CQR11] for the Airy$_2$ process.

View original: http://arxiv.org/abs/1201.4709